强非线性波动方程孤子行波解
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摘要
研究了一个强非线性波动方程.利用泛函分析变分迭代方法,首先构造了一个变分,求出相应的Lagrange乘子;其次构造一个解的变分迭代,选取初始孤子波;最后利用迭代方法依次求出各次孤子波的近似解.该方法是一个简单可行的近似求解非线性方程的方法.
A strongly nonlinear wave equation was studied. With the functional analytic variational iteration method,firstly,a variational iteration was constructed,and the corresponding Lagrangian multiplicator was solved. Secondly,the initial solitary wave was selected and the iteration method was used to obtain the approximate solution of arbitrary-degree accuracy for the solitary wave. This method is easy and feasible for getting approximate solutions to nonlinear wave equations.
A strongly nonlinear wave equation was studied. With the functional analytic variational iteration method,firstly,a variational iteration was constructed,and the corresponding Lagrangian multiplicator was solved. Secondly,the initial solitary wave was selected and the iteration method was used to obtain the approximate solution of arbitrary-degree accuracy for the solitary wave. This method is easy and feasible for getting approximate solutions to nonlinear wave equations.
引文
[1]MCPHADENMJ,ZHANG D.Slowdown of the meridional overturning circulation in the upper Pacific O cean[J].Nature,2002,415(6872):603-608.
[2]LOUTSENKO I.The variable coefficient Hele-Shaw problem,integrability and quadrature identities[J].Communications in Mathematical Physics,2006,268(2):465-479.
[3]GEDALINM.Low-frequency nonlinear stationary waves and fast shocks:hydrodynamical description[J].Physics of Plasmas,1998,5(1):127-132.
[4]PARKES E J.Some periodic and solitary travelling-wave solutions of the short-pulse equation[J].Chaos Solitons&Fractals,2008,38(1):154-159.
[5]GU Daifang,PHILANDER S G H.Interdecadal climate fluctuations that depend on exchanges between the tropics and extratropics[J].Science,1997,275(5301):805-807.
[6]马松华,强继业,方建平.(2+1)维Boiti-Leon-Pempinelli系统的混沌行为及孤子间的相互作用[J].物理学报,2007,56(2):620-626.(MA Songhua,QIANG Jiye,FANG Jianping.The interaction between solitons and chaotic behaviours of(2+1)-dimensional Boiti-Leon-Pempinelli system[J].A cta Physica Sinica,2007,56(2):620-626.(in Chinese))
[7]潘留仙,左伟明,颜家壬.Landau-Ginzburg-Higgs方程的微扰理论[J].物理学报,2005,54(1):1-5.(PANLiuxian,Z U O Weiming,YANJiaren.T he theory of the perturbation for LandauGinzburg-Higgs equation[J].A cta Physica Sinica,2005,54(1):1-5.(in Chinese))
[8]吴国将,韩家骅,史良马,等.一般变换下双Jacobi椭圆函数展开法及应用[J].物理学报,2006,55(8):3858-3863.(WU Guojiang,HANJiahua,SHI Liangma,et al.Double Jacobian elliptic function expansion method under a general function transform and its applications[J].A cta Physica Sinica,2006,55(8):3858-3863.(in Chinese))
[9]马松华,吴小红,方建平,等.(3+1)维Burgers系统的新精确解及其特殊孤立子结构[J].物理学报,2008,57(1):11-17.(MA Songhua,WU Xiaohong,FANG Jianping,et al.Newexact solutions and special soliton structures for the(3+1)-dimensional Burgers system[J].A cta Physica Sinica,2008,57(1):11-17.(in Chinese))
[10]李帮庆,马玉兰.(G'/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解[J].物理学报,2009,58(7):4373-4378.(LI Bangqing,MA Yulan.(G'/G)-expansion method and newexact solutions for(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov system[J].A cta Physica Sinica,2009,58(7):4373-4378.(in Chinese))
[11]周振春,马松华,方建平,等.(2+1)维孤子系统的多孤子解和分形结构[J].物理学报,2010,59(11):7540-7545.(Z HO U Z henchun,MA Songhua,FANG Jianping,et al.Multi-soliton solutions and fractal structures in a(2+1)-dimensional soliton system[J].A cta Physica Sinica,2010,59(11):7540-7545.(in Chinese))
[12]高亮,徐伟,唐亚宁,等.一类广义Boussinesq方程和Boussinesq-Burgers方程新的显式精确解[J].物理学报,2007,56(4):1860-1869.(GAO Liang,XU Wei,TANG Yaning,et al.Newexplicit exact solutions of one type of generalized Boussinesq equations and the Boussinesq-Burgers equation[J].A cta Physica Sinica,2007,56(4):1860-1869.(in Chinese))
[13]李向正,李修勇,赵丽英,等.Gerdjikov-Ivanov方程的精确解[J].物理学报,2008,57(4):2031-2034.(LING Xiangzheng,LI Xiuyong,Z HANG Liying,et al.Exact solutions of GerdjikovIvanov equation[J].A cta Physica Sinica,2008,57(4):2031-2034.(in Chinese))
[14]石兰芳,聂子文.应用全新G'/(G+G')展开方法求解广义非线性Schr9dinger方程和耦合非线性Schr9dinger方程组[J].应用数学和力学,2017,38(5):539-552.(SHI Lanfang,NIE Ziwen.Solutions to the nonlinear Schr9dinger equation and coupled nonlinear Schr9dinger equations with a new G'/(G+G')-expansion method[J].A pplied Mathematics and Mechanics,2017,38(5):539-552.(in Chinese))
[15]石兰芳,莫嘉琪.一类强非线性方程Robin问题奇摄动解[J].应用数学,2017,30(2):247-251.(SHI Lanfang,MO Jiaqi.A class of singular perturbation solutions to strong nonlinear equation R obin problems[J].Mathematica A pplicata,2017,30(2):247-251.(in Chinese))
[16]FENG Yihu,MO Jiaqi.Asymptotic solution for singularly perturbed fractional order differential equation[J].Journal of Mathematics,2016,36(2):239-245.
[17]FENG Yihu,CHENXianfeng,MO Jiaqi.The shock wave solution of a class of singularly perturbed problem for generalized nonlinear reaction diffusion equation[J].Mathematica A pplicata,2017,30(1):1-7.
[18]冯依虎,莫嘉琪.一类非线性非局部扰动LGH方程的孤子行波解[J].应用数学和力学,2016,37(4):426-433.(FENG Yihu,MO Jiaqi.Soliton travelling wave solutions to a class of nonlinear nonlocal disturbed LGH equations[J].A pplied Mathematics and Mechanics,2016,37(4):426-433.(in Chinese))
[19]冯依虎,石兰芳,莫嘉琪.关于将飞秒脉冲激光用于纳米金属薄膜传导系统的研究[J].工程数学学报,2017,34(1):13-20.(FENG Yihu,SHI Lanfang,MO Jiaqi.Study of transfers system for femtosecond pulse laser to nano metal film[J].Chinese Journal of Engineering Mathematics,2017,34(1):13-20.(in Chinese))
[20]冯依虎,陈怀军,莫嘉琪.一类非线性奇异摄动自治微分系统的渐近解[J].应用数学和力学,2017,38(5):355-363.(FENG Yihu,CHENHuaijun,MO Jiaqi.Asymptotic solution to a class of nonlinear singular perturbation autonomic differential system[J].A pplied Mathematics and Mechanics,2017,38(5):355-363.(in Chinese))
[21]冯依虎,林万涛,莫嘉琪.大气中尘埃扩散方程行波解[J].吉林大学学报(理学版),2016,54(2):234-240.(FENG Yihu,LINWantao,MO Jiaqi.Travelling wave solution for dust diffusion equation in atmosphere[J].Journal of Jilin University(Science Edition),2016,54(2):234-240.(in Chinese))
[22]冯依虎,林万涛,莫嘉琪.一类大气量子等离子流体动力学孤立子波渐近解[J].吉林大学学报(理学版),2017,55(3):474-480.(FENG Yihu,LINWantao,MO Jiaqi.Asymptotic solution for a class of quantum plasma fluid dynamics solitary wave in atmosphere[J].Journal of Jilin University(Science Edition),2017,55(3):474-480.(in Chinese))
[23]冯依虎,莫嘉琪.一类广义奇摄动非线性双曲型积分-微分方程模型[J].吉林大学学报(理学版),2017,55(5):1055-1060.(FENG Yihu,MO Jiaqi.A class of generalized nonlinear hyperbolic integral-differential equation with singular perturbation model[J].Journal of Jilin University(Science Edition),2017,55(5):1055-1060.(in Chinese))
[24]LEBEDEV L P,CLOUD MJ.The Calculus of Variations and Functional Analysis With Optimal Control and A pplications in Mechanics[M].New York:World Scientific,2003.
[25]何吉欢.工程和科学计算中的近似非线性分析方法[M].郑州:河南科学技术出版社,2002.(HE Jihuan.Approximate Nonlinear Analytical Methods in Engineering and Sciences[M].Zhengzhou:Henan Science and T echnology Press,2002.(in Chinese))
[26]DE JAGER M,JIANG F R.The Theory of Singular Perturbations[M].Amsterdam:North-Holland Publishing Co,1996.
[27]BARBU L,MOROSANU G.Singularly Perturbed Boundary-Value Problems[M].Basel:Birk-huser,2007.
[2]LOUTSENKO I.The variable coefficient Hele-Shaw problem,integrability and quadrature identities[J].Communications in Mathematical Physics,2006,268(2):465-479.
[3]GEDALINM.Low-frequency nonlinear stationary waves and fast shocks:hydrodynamical description[J].Physics of Plasmas,1998,5(1):127-132.
[4]PARKES E J.Some periodic and solitary travelling-wave solutions of the short-pulse equation[J].Chaos Solitons&Fractals,2008,38(1):154-159.
[5]GU Daifang,PHILANDER S G H.Interdecadal climate fluctuations that depend on exchanges between the tropics and extratropics[J].Science,1997,275(5301):805-807.
[6]马松华,强继业,方建平.(2+1)维Boiti-Leon-Pempinelli系统的混沌行为及孤子间的相互作用[J].物理学报,2007,56(2):620-626.(MA Songhua,QIANG Jiye,FANG Jianping.The interaction between solitons and chaotic behaviours of(2+1)-dimensional Boiti-Leon-Pempinelli system[J].A cta Physica Sinica,2007,56(2):620-626.(in Chinese))
[7]潘留仙,左伟明,颜家壬.Landau-Ginzburg-Higgs方程的微扰理论[J].物理学报,2005,54(1):1-5.(PANLiuxian,Z U O Weiming,YANJiaren.T he theory of the perturbation for LandauGinzburg-Higgs equation[J].A cta Physica Sinica,2005,54(1):1-5.(in Chinese))
[8]吴国将,韩家骅,史良马,等.一般变换下双Jacobi椭圆函数展开法及应用[J].物理学报,2006,55(8):3858-3863.(WU Guojiang,HANJiahua,SHI Liangma,et al.Double Jacobian elliptic function expansion method under a general function transform and its applications[J].A cta Physica Sinica,2006,55(8):3858-3863.(in Chinese))
[9]马松华,吴小红,方建平,等.(3+1)维Burgers系统的新精确解及其特殊孤立子结构[J].物理学报,2008,57(1):11-17.(MA Songhua,WU Xiaohong,FANG Jianping,et al.Newexact solutions and special soliton structures for the(3+1)-dimensional Burgers system[J].A cta Physica Sinica,2008,57(1):11-17.(in Chinese))
[10]李帮庆,马玉兰.(G'/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解[J].物理学报,2009,58(7):4373-4378.(LI Bangqing,MA Yulan.(G'/G)-expansion method and newexact solutions for(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov system[J].A cta Physica Sinica,2009,58(7):4373-4378.(in Chinese))
[11]周振春,马松华,方建平,等.(2+1)维孤子系统的多孤子解和分形结构[J].物理学报,2010,59(11):7540-7545.(Z HO U Z henchun,MA Songhua,FANG Jianping,et al.Multi-soliton solutions and fractal structures in a(2+1)-dimensional soliton system[J].A cta Physica Sinica,2010,59(11):7540-7545.(in Chinese))
[12]高亮,徐伟,唐亚宁,等.一类广义Boussinesq方程和Boussinesq-Burgers方程新的显式精确解[J].物理学报,2007,56(4):1860-1869.(GAO Liang,XU Wei,TANG Yaning,et al.Newexplicit exact solutions of one type of generalized Boussinesq equations and the Boussinesq-Burgers equation[J].A cta Physica Sinica,2007,56(4):1860-1869.(in Chinese))
[13]李向正,李修勇,赵丽英,等.Gerdjikov-Ivanov方程的精确解[J].物理学报,2008,57(4):2031-2034.(LING Xiangzheng,LI Xiuyong,Z HANG Liying,et al.Exact solutions of GerdjikovIvanov equation[J].A cta Physica Sinica,2008,57(4):2031-2034.(in Chinese))
[14]石兰芳,聂子文.应用全新G'/(G+G')展开方法求解广义非线性Schr9dinger方程和耦合非线性Schr9dinger方程组[J].应用数学和力学,2017,38(5):539-552.(SHI Lanfang,NIE Ziwen.Solutions to the nonlinear Schr9dinger equation and coupled nonlinear Schr9dinger equations with a new G'/(G+G')-expansion method[J].A pplied Mathematics and Mechanics,2017,38(5):539-552.(in Chinese))
[15]石兰芳,莫嘉琪.一类强非线性方程Robin问题奇摄动解[J].应用数学,2017,30(2):247-251.(SHI Lanfang,MO Jiaqi.A class of singular perturbation solutions to strong nonlinear equation R obin problems[J].Mathematica A pplicata,2017,30(2):247-251.(in Chinese))
[16]FENG Yihu,MO Jiaqi.Asymptotic solution for singularly perturbed fractional order differential equation[J].Journal of Mathematics,2016,36(2):239-245.
[17]FENG Yihu,CHENXianfeng,MO Jiaqi.The shock wave solution of a class of singularly perturbed problem for generalized nonlinear reaction diffusion equation[J].Mathematica A pplicata,2017,30(1):1-7.
[18]冯依虎,莫嘉琪.一类非线性非局部扰动LGH方程的孤子行波解[J].应用数学和力学,2016,37(4):426-433.(FENG Yihu,MO Jiaqi.Soliton travelling wave solutions to a class of nonlinear nonlocal disturbed LGH equations[J].A pplied Mathematics and Mechanics,2016,37(4):426-433.(in Chinese))
[19]冯依虎,石兰芳,莫嘉琪.关于将飞秒脉冲激光用于纳米金属薄膜传导系统的研究[J].工程数学学报,2017,34(1):13-20.(FENG Yihu,SHI Lanfang,MO Jiaqi.Study of transfers system for femtosecond pulse laser to nano metal film[J].Chinese Journal of Engineering Mathematics,2017,34(1):13-20.(in Chinese))
[20]冯依虎,陈怀军,莫嘉琪.一类非线性奇异摄动自治微分系统的渐近解[J].应用数学和力学,2017,38(5):355-363.(FENG Yihu,CHENHuaijun,MO Jiaqi.Asymptotic solution to a class of nonlinear singular perturbation autonomic differential system[J].A pplied Mathematics and Mechanics,2017,38(5):355-363.(in Chinese))
[21]冯依虎,林万涛,莫嘉琪.大气中尘埃扩散方程行波解[J].吉林大学学报(理学版),2016,54(2):234-240.(FENG Yihu,LINWantao,MO Jiaqi.Travelling wave solution for dust diffusion equation in atmosphere[J].Journal of Jilin University(Science Edition),2016,54(2):234-240.(in Chinese))
[22]冯依虎,林万涛,莫嘉琪.一类大气量子等离子流体动力学孤立子波渐近解[J].吉林大学学报(理学版),2017,55(3):474-480.(FENG Yihu,LINWantao,MO Jiaqi.Asymptotic solution for a class of quantum plasma fluid dynamics solitary wave in atmosphere[J].Journal of Jilin University(Science Edition),2017,55(3):474-480.(in Chinese))
[23]冯依虎,莫嘉琪.一类广义奇摄动非线性双曲型积分-微分方程模型[J].吉林大学学报(理学版),2017,55(5):1055-1060.(FENG Yihu,MO Jiaqi.A class of generalized nonlinear hyperbolic integral-differential equation with singular perturbation model[J].Journal of Jilin University(Science Edition),2017,55(5):1055-1060.(in Chinese))
[24]LEBEDEV L P,CLOUD MJ.The Calculus of Variations and Functional Analysis With Optimal Control and A pplications in Mechanics[M].New York:World Scientific,2003.
[25]何吉欢.工程和科学计算中的近似非线性分析方法[M].郑州:河南科学技术出版社,2002.(HE Jihuan.Approximate Nonlinear Analytical Methods in Engineering and Sciences[M].Zhengzhou:Henan Science and T echnology Press,2002.(in Chinese))
[26]DE JAGER M,JIANG F R.The Theory of Singular Perturbations[M].Amsterdam:North-Holland Publishing Co,1996.
[27]BARBU L,MOROSANU G.Singularly Perturbed Boundary-Value Problems[M].Basel:Birk-huser,2007.